A (combinatorial) embedding consists of cyclic orderings of the incident
edges at each vertex. Given such an embedding there are multiple approaches
discussed in literature to drawing the graph (subject to various
constraints, e.g. integer coordinates), see e.g. [2].

The planarity check algorithm and extraction of the combinatorial embedding
is based on the Left-Right Planarity Test [1].

A counterexample is only generated if the corresponding parameter is set,
because the complexity of the counterexample generation is higher.